Problem: $ E = \left[\begin{array}{rr}3 & 5 \\ -1 & 1\end{array}\right]$ $ A = \left[\begin{array}{rrr}-2 & 2 & 3 \\ 3 & 5 & -2\end{array}\right]$ What is $ E A$ ?
Because $ E$ has dimensions $(2\times2)$ and $ A$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ E A = \left[\begin{array}{rr}{3} & {5} \\ {-1} & {1}\end{array}\right] \left[\begin{array}{rrr}{-2} & \color{#DF0030}{2} & \color{#9D38BD}{3} \\ {3} & \color{#DF0030}{5} & \color{#9D38BD}{-2}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ E$ , with the corresponding elements in column $j$ of the second matrix, $ A$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ E$ with the first element in ${\text{column }1}$ of $ A$ , then multiply the second element in ${\text{row }1}$ of $ E$ with the second element in ${\text{column }1}$ of $ A$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{3}\cdot{-2}+{5}\cdot{3} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ E$ with the corresponding elements in ${\text{column }1}$ of $ A$ and add the products together. $ \left[\begin{array}{rrr}{3}\cdot{-2}+{5}\cdot{3} & ? & ? \\ {-1}\cdot{-2}+{1}\cdot{3} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ E$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ A$ and add the products together. $ \left[\begin{array}{rrr}{3}\cdot{-2}+{5}\cdot{3} & {3}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{5} & ? \\ {-1}\cdot{-2}+{1}\cdot{3} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{3}\cdot{-2}+{5}\cdot{3} & {3}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{5} & {3}\cdot\color{#9D38BD}{3}+{5}\cdot\color{#9D38BD}{-2} \\ {-1}\cdot{-2}+{1}\cdot{3} & {-1}\cdot\color{#DF0030}{2}+{1}\cdot\color{#DF0030}{5} & {-1}\cdot\color{#9D38BD}{3}+{1}\cdot\color{#9D38BD}{-2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}9 & 31 & -1 \\ 5 & 3 & -5\end{array}\right] $